Select The Correct Answer.What Is The Value Of $x$? Log ⁡ 2 ( X + 7 ) = 2 \log_2(x+7) = 2 Lo G 2 ​ ( X + 7 ) = 2 A. -3 B. 11 C. 22 D. 250

Introduction

Logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving a specific type of logarithmic equation, namely $\log_2(x+7) = 2$. We will break down the solution into manageable steps, making it easy to understand and follow along.

Understanding Logarithmic Equations

Before we dive into the solution, let's take a moment to understand what logarithmic equations are. A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, if $y = 2^x$, then $x = \log_2(y)$. Logarithmic equations can be solved using various techniques, including algebraic manipulation and logarithmic properties.

Step 1: Rewrite the Equation

The given equation is $\log_2(x+7) = 2$. To solve for $x$, we need to rewrite the equation in exponential form. Using the definition of logarithms, we can rewrite the equation as:

$$2^2 = x + 7$$

Step 2: Simplify the Equation

Now that we have rewritten the equation in exponential form, we can simplify it by evaluating the left-hand side:

$$4 = x + 7$$

Step 3: Isolate the Variable

To solve for $x$, we need to isolate the variable on one side of the equation. We can do this by subtracting 7 from both sides:

$$-3 = x$$

Conclusion

And there you have it! We have solved the logarithmic equation $\log_2(x+7) = 2$ and found that the value of $x$ is $-3$. This is the correct answer, and we can verify it by plugging it back into the original equation.

Why Choose the Correct Answer?

When solving logarithmic equations, it's essential to choose the correct answer. In this case, the correct answer is $-3$. The other options, $11$, $22$, and $250$, are not correct solutions to the equation.

Tips and Tricks

Here are some tips and tricks to help you solve logarithmic equations like this one:

• Understand the properties of logarithms: Logarithmic equations involve logarithmic properties, such as the product rule and the power rule. Make sure you understand these properties before attempting to solve a logarithmic equation.
• Use algebraic manipulation: Logarithmic equations can be solved using algebraic manipulation. Look for opportunities to simplify the equation and isolate the variable.
• Check your work: Once you think you have found the solution, plug it back into the original equation to verify that it is correct.

Common Mistakes to Avoid

When solving logarithmic equations, there are several common mistakes to avoid:

• Not understanding the properties of logarithms: Logarithmic equations involve logarithmic properties, such as the product rule and the power rule. Make sure you understand these properties before attempting to solve a logarithmic equation.
• Not using algebraic manipulation: Logarithmic equations can be solved using algebraic manipulation. Look for opportunities to simplify the equation and isolate the variable.
• Not checking your work: Once you think you have found the solution, plug it back into the original equation to verify that it is correct.

Conclusion

Solving logarithmic equations can be challenging, but with the right approach, they can be tackled with ease. By understanding the properties of logarithms, using algebraic manipulation, and checking your work, you can solve logarithmic equations like $\log_2(x+7) = 2$ and find the correct value of $x$. Remember to avoid common mistakes, such as not understanding the properties of logarithms, not using algebraic manipulation, and not checking your work. With practice and patience, you can become proficient in solving logarithmic equations and tackle even the most challenging problems.

Final Answer

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Introduction

Logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving a specific type of logarithmic equation, namely $\log_2(x+7) = 2$. We will break down the solution into manageable steps, making it easy to understand and follow along.

Q&A: Logarithmic Equations

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, if $y = 2^x$, then $x = \log_2(y)$.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to rewrite the equation in exponential form and then use algebraic manipulation to isolate the variable.

Q: What are some common mistakes to avoid when solving logarithmic equations?

A: Some common mistakes to avoid when solving logarithmic equations include not understanding the properties of logarithms, not using algebraic manipulation, and not checking your work.

Q: How do I choose the correct answer when solving a logarithmic equation?

A: When solving a logarithmic equation, you need to choose the correct answer by plugging it back into the original equation and verifying that it is correct.

Q: What are some tips and tricks for solving logarithmic equations?

A: Some tips and tricks for solving logarithmic equations include understanding the properties of logarithms, using algebraic manipulation, and checking your work.

Q: Can you provide an example of a logarithmic equation?

A: Yes, here is an example of a logarithmic equation: $\log_2(x+7) = 2$. We can solve this equation by rewriting it in exponential form and then using algebraic manipulation to isolate the variable.

Q: How do I rewrite a logarithmic equation in exponential form?

A: To rewrite a logarithmic equation in exponential form, you need to use the definition of logarithms. For example, if $\log_2(x+7) = 2$, then $2^2 = x + 7$.

Q: Can you provide a step-by-step solution to the logarithmic equation $\log_2(x+7) = 2$?

A: Yes, here is a step-by-step solution to the logarithmic equation $\log_2(x+7) = 2$:

1. Rewrite the equation in exponential form: $2^2 = x + 7$
2. Simplify the equation: $4 = x + 7$
3. Isolate the variable: $-3 = x$

Q: What is the final answer to the logarithmic equation $\log_2(x+7) = 2$?

A: The final answer to the logarithmic equation $\log_2(x+7) = 2$ is $\boxed{-3}$.

Conclusion

Solving logarithmic equations can be challenging, but with the right approach, they can be tackled with ease. By understanding the properties of logarithms, using algebraic manipulation, and checking your work, you can solve logarithmic equations like $\log_2(x+7) = 2$ and find the correct value of $x$. Remember to avoid common mistakes, such as not understanding the properties of logarithms, not using algebraic manipulation, and not checking your work. With practice and patience, you can become proficient in solving logarithmic equations and tackle even the most challenging problems.

Final Answer

The final answer is $\boxed{-3}$.